J. Birman :
“Studying links via closed braids ,”
pp. 1–67
in
Lecture notes of the ninth KAIST mathematics workshop
(Taejon, Korea, 1–13 August 1994 ),
vol. 1 .
Edited by S. H. Bae, G. T. Jin, and K. H. Ko .
Korea Advanced Institute of Science and Technology (Taejon ),
1994 .
Zbl
0835.57002
incollection

People
BibTeX
@incollection {key0835.57002z,
AUTHOR = {Birman, Joan},
TITLE = {Studying links via closed braids},
BOOKTITLE = {Lecture notes of the ninth {KAIST} mathematics
workshop},
EDITOR = {Bae, S. H. and Jin, G. T. and Ko, K.
H.},
VOLUME = {1},
PUBLISHER = {Korea Advanced Institute of Science
and Technology},
ADDRESS = {Taejon},
YEAR = {1994},
PAGES = {1--67},
NOTE = {(Taejon, Korea, 1--13 August 1994).
Zbl:0835.57002.},
}
J. Birman, K. H. Ko, and S. J. Lee :
“A new approach to the word and conjugacy problems in the braid groups ,”
Adv. Math.
139 : 2
(November 1998 ),
pp. 322–353 .
MR
1654165
Zbl
0937.20016
article

Abstract
People
BibTeX

A new presentation of the \( n \) -string braid group \( B_n \) is studied. Using it, a new solution to the word problem in \( B_n \) is obtained which retains most of the desirable features of the Garside–Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.

@article {key1654165m,
AUTHOR = {Birman, Joan and Ko, Ki Hyoung and Lee,
Sang Jin},
TITLE = {A new approach to the word and conjugacy
problems in the braid groups},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {139},
NUMBER = {2},
MONTH = {November},
YEAR = {1998},
PAGES = {322--353},
DOI = {10.1006/aima.1998.1761},
NOTE = {MR:1654165. Zbl:0937.20016.},
ISSN = {0001-8708},
CODEN = {ADMTA4},
}
J. S. Birman, K. H. Ko, and S. J. Lee :
“The infimum, supremum, and geodesic length of a braid conjugacy class ,”
Adv. Math.
164 : 1
(December 2001 ),
pp. 41–56 .
MR
1870512
Zbl
1063.20039
article

Abstract
People
BibTeX

Algorithmic solutions to the conjugacy problem in the braid groups \( B_n \) , \( n = 2,\,3,\,4,\dots \) were given in earlier work. This xlinconcerns the computation of two integer class invariants, known as “inf” and “sup.” A key issue in both algorithms is the number \( m \) of times one must “cycle” (resp. “decycle”) in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that \( m \) is bounded above by
\[ \frac{n^2-n}{2}-1 \]
in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford 45 , 479–497) and by \( n-2 \) in the situation stated by authors (1998, Adv. Math. 139 , 322–353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length.

@article {key1870512m,
AUTHOR = {Birman, Joan S. and Ko, Ki Hyoung and
Lee, Sang Jin},
TITLE = {The infimum, supremum, and geodesic
length of a braid conjugacy class},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {164},
NUMBER = {1},
MONTH = {December},
YEAR = {2001},
PAGES = {41--56},
DOI = {10.1006/aima.2001.2010},
NOTE = {MR:1870512. Zbl:1063.20039.},
ISSN = {0001-8708},
CODEN = {ADMTA4},
}
J. S. Birman and W. W. Menasco :
“On Markov’s theorem ,”
pp. 295–310
in
Knots 2000 Korea (Volume 1)
(Yongpyong, Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 3 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
2002 .
MR
1905686
Zbl
1059.57002
incollection

Abstract
People
BibTeX

Let \( \chi \) be an oriented link type in the oriented 3-sphere \( S^3 \) or
\[ \mathbb{R}^3 = S^3 - \{\infty\} .\]
A representative \( X \in \chi \) is said to be a closed braid if there is an unknotted curve
\[ \mathbf{A} \subset S^3 - X \]
(the axis ) and a choice of fibration \( \mathscr{H} \) of the open solid torus \( S^3 - \mathbf{A} \) by meridian discs
\[ \{H_{\theta}: \theta\in [0, 2\pi]\} ,\]
such that whenever \( X \) meets a fiber \( H_{\theta} \) the intersection is transverse.

Closed braid representations of \( \chi \) are not unique, and Markov’s well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov’s theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.

@article {key1905686m,
AUTHOR = {Birman, Joan S. and Menasco, William
W.},
TITLE = {On {M}arkov's theorem},
JOURNAL = {J. Knot Theor. Ramif.},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {11},
NUMBER = {3},
YEAR = {2002},
PAGES = {295--310},
DOI = {10.1142/S0218216502001627},
NOTE = {\textit{Knots 2000 Korea (Volume 1)}
(Yongpyong, Korea, 31 July--5 August
2000). Issue edited by J. S. Birman,
C. M. Gordon, G. T. Jin,
L. H. Kauffman, A. Kawauchi,
K. H. Ko, J. P. Levine,
and Y. Matsumoto. MR:1905686.
Zbl:1059.57002.},
ISSN = {0218-2165},
}
J. S. Birman, M. Rampichini, P. Boldi, and S. Vigna :
“Towards an implementation of the B–H algorithm for recognizing the unknot ,”
pp. 601–645
in
Knots 2000 Korea (Volume 2)
(Yongpyong, Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 4 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Hackensack, NJ ),
2002 .
MR
1915499
Zbl
1007.57004
incollection

Abstract
People
BibTeX

In the manuscript [Birman and Hirsch 1998] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a ‘braid foliation’ and are embeddable in 3-space. The boundaries of these ‘foliated embeddable discs’ (FEDs) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FEDs. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list.

In this paper we describe implementations of the first and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FEDs have unexplored and interesting structure. Open question are interspersed throughout the manuscript.

The third part of the algorithm was studied in [Birman, Ko and Lee 1998] and [Birman, Ko and Lee 2001], and implemented by S. J. Lee [preprint]. At this writing his algorithm is polynomial for \( n\leq 4 \) and exponential for \( n\geq 5 \) .

@article {key1915499m,
AUTHOR = {Birman, Joan S. and Rampichini, Marta
and Boldi, Paolo and Vigna, Sebastiano},
TITLE = {Towards an implementation of the {B}--{H}
algorithm for recognizing the unknot},
JOURNAL = {J. Knot Theor. Ramif.},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {11},
NUMBER = {4},
YEAR = {2002},
PAGES = {601--645},
DOI = {10.1142/S0218216502001858},
NOTE = {\textit{Knots 2000 Korea (Volume 2)}
(Yongpyong, Korea, 31 July--5 August
2000). Issue edited by J. S. Birman,
C. M. Gordon, G. T. Jin,
L. H. Kauffman, A. Kawauchi,
K. H. Ko, J. P. Levine,
and Y. Matsumoto. MR:1915499.
Zbl:1007.57004.},
ISSN = {0218-2165},
}
Knots 2000 Korea (Volume 1)
(Yongpyong, South Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 3 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
2002 .
Zbl
0995.00510
book

People
BibTeX
@book {key0995.00510z,
TITLE = {Knots 2000 Korea (Volume 1)},
EDITOR = {Birman, J. S. and Gordon, C. M. and
Jin, G. T. and Kauffman, L. H. and Kawauchi,
A. and Ko, K. H. and Levine, J. P. and
Matsumoto, Y.},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
YEAR = {2002},
PAGES = {283--473},
URL = {http://www.worldscientific.com/toc/jktr/11/03},
NOTE = {(Yongpyong, South Korea, 31 July--5
August 2000). Published as \textit{J.
Knot Theor. Ramif.} \textbf{11}:3. Zbl:0995.00510.},
ISSN = {0218-2165},
}
Knots 2000 Korea (Volume 3)
(Yongpyong, South Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 6 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
September 2002 .
Zbl
1018.00508
book

People
BibTeX
@book {key1018.00508z,
TITLE = {Knots 2000 Korea (Volume 3)},
EDITOR = {Birman, J. S. and Gordon, C. M. and
Jin, G. T. and Kauffman, L. H. and Kawauchi,
A. and Ko, K. H. and Levine, J. P. and
Matsumoto, Y.},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
MONTH = {September},
YEAR = {2002},
PAGES = {833--1016},
URL = {http://www.worldscientific.com/toc/jktr/11/06},
NOTE = {(Yongpyong, South Korea, 31 July--5
August 2000). Published as \textit{J.
Knot Theor. Ramif.} \textbf{11}:6. Zbl:1018.00508.},
ISSN = {0218-2165},
}